Abstract
Consider a class of graphs \(\mathcal{G}\) having a polynomial time algorithm computing the set of all minimal separators for every graph in \(\mathcal{G}\). We show that there is a polynomial time algorithm for treewidth and minimum fill-in, respectively, when restricted to the class \(\mathcal{G}\). Many interesting classes of intersection graphs have a polynomial time algorithm computing all minimal separators, like permutation graphs, circle graphs, circular arc graphs, distance hereditary graphs, chordal bipartite graphs etc. Our result generalizes earlier results for the treewidth and minimum fill-in for several of these classes. We also consider the related problems pathwidth and interval completion when restricted to some special graph classes.
The work of the first and second author has been supported partially by the ESPRIT Basic Research Action of the EC under contract No. 7141 (project ALCOM II.)
This author is supported by the Foundation for Computer Science (S.I.O.N.) of the Netherlands Organization for Scientific Research (N.W.O.).
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References
Anand, R., H. Balakrishnan and C. Pandu Rangan, Treewidth of distance hereditary graphs, To appear.
Arnborg, S., Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey. BIT 25, (1985), pp. 2–23.
Arnborg, S., D. G. Corneil and A. Proskurowski, Complexity of finding embeddings in a k-tree, SIAM J. Alg. Disc. Meth. 8, (1987), pp. 277–284.
Arnborg, S. and A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees. Disc. Appl. Math. 23, (1989), pp. 11–24.
Bandelt, H. J. and H. M. Mulder, Distance-Hereditary Graphs, Journal of Combinatorial Theory, Series B 41, (1986), pp. 182–208.
Bodlaender, H., Classes of graphs with bounded treewidth, Technical Report RUU-CS-86-22, Department of Computer Science, Utrecht University, 1986.
Bodlaender, H., A tourist guide through treewidth, Technical report RUU-CS-92-12, Department of Computer Science, Utrecht University, Utrecht, The Netherlands, 1992. To appear in: Acta Cybernetica.
Bodlaender, H., A linear time algorithm for finding tree-decompositions of small treewidth, In Proceedings of the 25th Annual ACM Symposium on Theory of Computing, 1993, pp. 226–234.
Bodlaender, H., T. Kloks and D. Kratsch, Treewidth and pathwidth of permutation graphs, Technical report RUU-CS-92-30, Department of Computer Science, Utrecht University, Utrecht, The Netherlands, (1992). To appear in: Proceedings of the 20 th International Colloquium on Automata, Languages and Programming (1993).
Bodlaender, H. and R. H. Möhring, The pathwidth and treewidth of cographs, In Proceedings 2nd Scandinavian Workshop on Algorithm Theory, Springer Verlag, Lecture Notes in Computer Science 447, (1990), pp. 301–309.
Brandstädt, A., Special graph classes — a survey, Schriftenreihe des Fachbereichs Mathematik, SM-DU-199 (1991), Universität-Gesamthochschule Duisburg.
Corneil, D. G., Y. Perl, L. K. Stewart, Cographs: recognition, applications and algorithms, Congressus Numerantium, 43 (1984), pp. 249–258.
Dirac, G. A., On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25, (1961), pp. 71–76.
Garey, M. R. and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, San Francisco, 1979.
Golumbic, M. C., Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
Golumbic, M. C., D. Rotem, J. Urrutia, Comparability graphs and intersection graphs, Discrete Mathematics 43, (1983), pp. 37–46.
Gustedt, J., On the pathwidth of chordal graphs, To appear in Discrete Applied Mathematics.
Habib, M. and R. H. Möhring, Treewidth of cocomparability graphs and a new order-theoretic parameter, Technical Report 336/1992, Technische Universität Berlin, 1992.
Johnson, D. S., The NP-completeness column: An ongoing guide, J. Algorithms 6, (1985), pp. 434–451.
Kloks, T., Treewidth, Ph.D. Thesis, Utrecht University, The Netherlands, 1993.
Kloks, T., Minimum fill-in for chordal bipartite graphs, Technical Report RUU-CS-93-11, Department of Computer Science, Utrecht University, 1993.
Kloks, T., Treewidth of circle graphs, Technical Report RUU-CS-93-12, Department of Computer Science, Utrecht University, 1993.
Kloks, T. and D. Kratsch, Treewidth of chordal bipartite graphs, 10 th Annual Symposium on Theoretical Aspects of Computer Science, Springer-Verlag, Lecture Notes in Computer Science 665, (1993), pp. 80–89.
Möhring, R. H., private communication.
Monien, B. and I. H. Sudborough, Min Cut is NP-complete for Edge Weighted Trees, Theoretical Computer Science 58 (1988), pp. 209–229.
Rose, D. J., Triangulated graphs and the elimination process, J. Math. Anal. Appl., 32 (1970), pp. 597–609.
Spinrad, J., A. Brandstädt, L. Stewart, Bipartite permutation graphs, Discrete Applied Mathematics, 18 (1987), pp. 279–292.
Sundaram, R., K. Sher Singh and C. Pandu Rangan, Treewidth of circular arc graphs, To appear in SIAM Journal on Discrete Mathematics.
Tarjan, R. E., Decomposition by clique separators, Discrete Mathematics 55 (1985), pp. 221–232.
Yannakakis, M., Computing the minimum fill-in is NP-complete, SIAM J. Alg. Disc. Meth. 2, (1981), pp. 77–79.
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Kloks, T., Bodlaender, H., Müller, H., Kratsch, D. (1993). Computing treewidth and minimum fill-in: All you need are the minimal separators. In: Lengauer, T. (eds) Algorithms—ESA '93. ESA 1993. Lecture Notes in Computer Science, vol 726. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57273-2_61
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DOI: https://doi.org/10.1007/3-540-57273-2_61
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